Femtosecond electron microscopy of relativistic electron bunches

The development of plasma-based accelerators has enabled the generation of very high brightness electron bunches of femtosecond duration, micrometer size and ultralow emittance, crucial for emerging applications including ultrafast detection in material science, laboratory-scale free-electron lasers and compact colliders for high-energy physics. The precise characterization of the initial bunch parameters is critical to the ability to manipulate the beam properties for downstream applications. Proper diagnostic of such ultra-short and high charge density laser-plasma accelerated bunches, however, remains very challenging. Here we address this challenge with a novel technique we name as femtosecond ultrarelativistic electron microscopy, which utilizes an electron bunch from another laser-plasma accelerator as a probe. In contrast to conventional microscopy of using very low-energy electrons, the femtosecond duration and high electron energy of such a probe beam enable it to capture the ultra-intense space-charge fields of the investigated bunch and to reconstruct the charge distribution with very high spatiotemporal resolution, all in a single shot. In the experiment presented here we have used this technique to study the shape of a laser-plasma accelerated electron beam, its asymmetry due to the drive laser polarization, and its beam evolution as it exits the plasma. We anticipate that this method will significantly advance the understanding of complex beam-plasma dynamics and will also provide a powerful new tool for real-time optimization of plasma accelerators.

ultra-thin charged disk moving at light speed, the plasma density of the linear wake has a simple form close to sin(k p (z − ct))G(x)H(y) [2,3], where k p is the plasma wave number and G(x)H(y) resembles the bi-Gaussian lateral profile of the driver. As a result, Eq. S1 is further transformed into, Here we define the start point of the wakefield (the driver position) as z = 0 and no wakefield at z > 0. ∆ x is the horizontal RMS size of the driver and ∆ρ 0 is the density perturbation amplitude of the wakefield. k p , ∆ x and ∆ρ 0 can thus be simply estimated based on Eq. S2 with the probed wakefield images.
We now consider the probe electrons that traverse the driver. Since they do not pass through the full extent of the wakefield, the corresponding density modulation is slightly different from Eq. S1, I wake (y, z) κ +∞ z ρ(z − x, x, y)dx. This integral can be numerically solved and the obtained I wake (y, z) for z > 0 (preceding the plasma wake) has a similar form as the bi-Gaussian function G(z)H(y) (Though the distribution along z is slightly different).
Given that the contribution of wakefield to the probe deflection is still much less than that of the beam self-fields, we simplify the above relation as, where σ w0 = +∞ 0

MODULATION
When an ultra-relativistic electron bunch propagates in a plasma (pre-ionized by the residual laser in our case), it pushes outwards the plasma electrons that form the trailing wakefield. For a typical LPA electron bunch, its pulse duration (i.e., a few femtoseconds) is usually shorter than the plasma wavelength (10s µm for density below 10 18 cm −3 ). The plasma electrons thus experience an instantaneous impulse in this transverse direction. For the driver's charge of tens of pC, this impulse can be very intense such that the electrons are accelerated up to relativistic speeds. As a consequence, lateral plasma currents increase rapidly within the drive bunch, which produce transverse electromagnetic fields against the space-field fields from the bunch. This plasma screening phenomenon is observed in our PIC simulation when comparing the transverse E fields before and after the bunch enters the plasma (see Fig. S4 and section III), and was partially discussed in Ref [2,4], whereas no clear analytical model was presented to estimate the relevant fields. It is noted that the bunch because of divergence also contains radial currents that could introduce similar effects as the plasma currents. However the generated fields over the main beam fields are on the same order of the bunch divergence (< 10 −2 ) and thus this effect is neglected here for simplicity.
In the following, we derive a simple model for the plasma screening to an ultra-short bunch based on the Maxwell equations and linear wakefield theory. In the Lorenz gauge, the potential equations are, where µ 0 is the vacuum permeability, A and φ represent the vector and scalar potentials, respectively, ρ p and ρ b represent the plasma and beam charge densities, respectively, J p and J b represent the plasma and beam current densities respectively. In the quasi-static approximation, the coordinates (x, y, z, t) are substituted with (x, y, ξ = ct − z) based on the assumption of non-evolving driver. The above equations then become, where ψ = φ − cA z . Since the drive bunch is very short, it is reasonable to assume over the small bunch length, that the plasma electrons only gain a lateral momentum with no longitudinal motion, and that the plasma density is not perturbed, i.e., J p,z ≈ 0 and ρ p ≈ 0.
Neglecting the bunch's transverse divergence, it is then straightforward to conclude that, in this region, φ results from the bunch's charge density ρ b , A z from the bunch's longitudinal current density J b,z , and A ⊥ from the plasma lateral current density J p,⊥ . Transverse E ⊥ and B ⊥ are then calculated from the potentials as, where the first terms on the right hand of both formulas defined as E b,⊥ and B b,⊥ come from the drive bunch and the second terms defined as E p,⊥ and B p,⊥ come from the plasma lateral currents.
From the PIC simulation, it is shown that the shape of E p,⊥ is close to the bunch selffields both longitudinally and transversely. Assuming the drive bunch 3D distribution as a tri-Gaussian shape and Differentiating Eq. S6 by ξ, one can obtain the peak value of E p,⊥ as E p,⊥0 = 2/π(k p δ z )E z0 , where δ z is the bunch longitudinal RMS size and E z0 is the maximum plasma longitudinal field. Since E z0 can be easily calculated from ∆ρ 0 in linear wakefield theory. As a result, the ratio between E p,⊥ and E b,⊥ is expressed as, where ∆ x and ∆ y represent the bunch's transverse RMS sizes, σ b0 is the bunch's peak surface charge density and χ = Using the same derivation as Eq. 1 of the mansucript, the probe density modulation caused by the plasma screening is then, where G(z) and H(y) are defined the same as the previous section, and σ s0 = 2πk p χ∆ρ 0 is obtained directly from Eq. S7.
Taking these two effects into consideration, the total probe density modulation can thus be simply expressed as, where σ w0 and σ s0 can be obtained from Eq. S3 and Eq. S8 respectively, and σ b0 represents the bunch peak surface charge density and can be obtained from Eq. 1 of the manuscript.
Since it is apparent that σ b0 is negative and both σ w0 and σ s0 are positive, the bunch charge would be underestimated without considering these two effects. Eq. 2 of the manuscript is a general form of Eq. S9.

III. 3D PIC SIMULATION COMPARISON WITH THE EXPERIMENT
From the experimental probe image of Fig. 2a of the manuscript, we can calculate the electron bunch's lateral distribution. Fitting the elliptical structure with a bi-Gaussian profile, the RMS sizes of the vertical and horizontal are estimated as ∆ y = 7.5 µm and ∆ x = 12.5 µm respectively. For estimating the beam charge, we need to take into account the above two effects. First, from the plasma wakefield image captured by the probe, the plasma density perturbation can be obtained based on Eq. S2. Afterwards, it is straightforward to get the value of σ w0 as 18 ± 2 fC µm −2 based on Eq. S3. Meanwhile, σ s0 can be calculated as 30 ± 3 fC µm −2 from the plasma screening formula Eq. S8. As a result, the peak surface charge density of the bunch σ b0 is obtained as −160 ± 17 fC µm −2 from Eq. S9. The overall charge of the investigated electron bunch is thus calculated as 94 ± 10 pC with the error bar arising from the probe peak energy fluctuation and the uncertainties of probe/probed electron beams duration (see Methods and Materials).
To directly compare with the above experimental data, we have performed a 3D PIC simulation using FBPIC [5] with solely an electron bunch propagating in a plasma of the density 2.5 × 10 16 cm −3 . The reason why the laser was not included is that a separate PIC simulation showed that the laser intensity at this position was already quite weak such that the generated wakefield strength was only one eighth of the beam driven wakefield.
The bunch was set to have a tri-Gaussian distribution with the RMS sizes as ∆ x = 12.5 µm (horizontal), ∆ y = 7.5 µm (vertical) and δ z = 1 µm (longitudinal) respectively. The energy spectrum contained a peak energy of 150 MeV with 45% FWHM spread (the same as experiment) and the overall charge was 94 pC. For simplicity, the bunch was set with no transverse divergence. More details about the simulation setup can be found in the Methods and Materials.
After propagating for around 300 µm in plasma, the electron bunch launches a stable quasi-linear plasma wave, where the on-axis electron density is around 7 × 10 15 cm −3 (see Fig. S3). In Fig S4, we also compare the E y field before and after the electron bunch enters the plasma, and one can see the E x within the bunch is reduced by 12% (see Fig. S4c).
At this moment, an electron probe (the same parameters as Fig.2 of the manuscript) traverses the electron bunch and the associated wakefield along the x axis. After drifting 13 mm, the generated density modulation is plotted in Fig. S5a. Fitting the elliptical hollow structure with a bi-Gaussian function, we obtain the bunch's transverse RMS sizes as ∆ x = 13.9 µm and ∆ y = 8.4 µm respectively, which are slightly larger than the actual beam sizes by 12% because some of the probe electrons start to cross other's traces. Meanwhile, the estimated peak surface charge density of the bunch is slightly lower than the experi-mental one. As a result, the overall charge based on the analytical model gives 98 pC, in good agreement with the input value of 94 pC. Without considering the plasma effects, the estimated charge is lower by 29%.
It is noted that with even longer drift distance (stronger density modulation) the estimated total charge increases very slightly, whereas the sizes of elliptical structures become larger due to the trajectory expansion of the probe particles. Therefore, in order to get a reasonable size measurement (deviation less than 20%), the probe relative density modulation is better to satisfy |I total | < 0.5.